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Binary Numbers

The number system we usually use (the one you probably learned in elementary school) is called base 10, because each digit has ten possible values (1, 2, 3, 4, 5, 6, 7, 8, 9, and 0).

But computers don't have digits with ten possible values. They have bits with two possible values (0 and 1). So they use base 2 numbers.

Base 10 is also called decimal. Base 2 is also called binary.

Decimal, or "base-ten" numbers

To understand binary, let's take a closer look at how decimal numbers work. Take the number "101" in decimal:

In base 10, the digits 101 represent 1 hundred, 0 tens, and 1 one.

Notice we have two "1"s here, but they don't mean the same thing. The leftmost "1" means 100, and the rightmost "1" means 1. That's because the leftmost "1" is in the hundreds place, while the rightmost "1" is in the ones place. And the "0" between them is in the tens place.

In base 10, the digits 101 represent 1 hundred, 0 tens, and 1 one.

So this "101" in base 10 is telling us we have "1 hundred, 0 tens, and 1 one."

In base 10, the digits 101 represent 1 hundred, 0 tens, and 1 one, which add to give the value one hundred and one.

Notice how the places in base 10 (ones place, tens place, hundreds place, etc.) are sequential powers of 10:

  • 10^0=1
  • 10^1=10
  • 10^2=100
  • 10^3=1000
  • etc.

Binary, or "base-two" numbers

Just as the places in base 10 are sequential powers of 10, the places in binary (base 2) are sequential powers of 2:

  • 2^0=1
  • 2^1=2
  • 2^2=4
  • 2^3=8
  • etc.

So let's take that same "101" but this time let's read it as a binary number:

In base 2, the digits 101 represent 1 four, 0 twos, and 1 one.

Reading this from right to left: we have a 1 in the ones place, a 0 in the twos place, and a 1 in the fours place. So our total is 4 + 0 + 1 which is 5.

In base 2, the digits 101 represent 1 four, 0 twos, and 1 one, which add to give the value five.

Counting to 10 in binary

Here's how we'd count from 0 to 10 in binary, along with how we'd interpret each bit:

Decimal Binary Interpretation
0 0000 0 + 0 + 0 + 0
1 0001 0 + 0 + 0 + 1
2 0010 0 + 0 + 2 + 0
3 0011 0 + 0 + 2 + 1
4 0100 0 + 4 + 0 + 0
5 0101 0 + 4 + 0 + 1
6 0110 0 + 4 + 2 + 0
7 0111 0 + 4 + 2 + 1
8 1000 8 + 0 + 0 + 0
9 1001 8 + 0 + 0 + 1
10 1010 8 + 0 + 2 + 0

Some languages spoken in Nigeria and India use duodecimal numbers, or base-12. So "eleven" and "twelve" aren't built using 1s and 2s, they're entirely different digits. Some mathematicians argue that base-12 is a better system than our base-10, because 12 has more factors (1, 2, 3, 4, 6) than 10 does (1, 2, 5). We probably use decimal numbers because we have 10 fingers.

Negative Numbers and Two's Complement

Negative numbers are typically represented in binary using two's complement encoding. In two's complement, the leftmost digit is negative, and the rest of the digits are positive.

Let's look at what happens when we interpret that 101 as two's complement:

When interpreting binary values in 2s complement, the digits 101 represent 1 negative four, 0 twos, and 1 one, which add to give the value negative three.

Fun computer systems trivia fact: two's complement isn't the only way negative numbers could be encoded. Other encodings tossed around in the 1960s included one's complement and sign and magnitude encodings. Of the three encodings, two's complement is the one still used today for a few reasons:

  1. There is only one way to represent zero.
  2. Basic operations like addition, subtraction, and multiplication are the same regardless of whether the numbers involved are positive or negative.

Since two's complement had both of these properties (and the others didn't), it stuck around and is still used today.

Counting from -5 to 5 in two's complement

Here are the base-10 numbers -5 through 5 in two's complement, along with how we'd interpret each bit:

Decimal Binary Interpretation
-5 1011 -8 + 0 + 2 + 1
-4 1100 -8 + 4 + 0 + 0
-3 1101 -8 + 4 + 0 + 1
-2 1110 -8 + 4 + 2 + 0
-1 1111 -8 + 4 + 2 + 1
0 0000 0 + 0 + 0 + 0
1 0001 0 + 0 + 0 + 1
2 0010 0 + 0 + 2 + 0
3 0011 0 + 0 + 2 + 1
4 0100 0 + 4 + 0 + 0
5 0101 0 + 4 + 0 + 1

So, should 1011 be read as "eleven" (in binary) or "negative five" (in two's complement)?

It could be either one! Many languages have two types of numbers: signed and unsigned. Signed numbers are represented in two's complement, and unsigned numbers use plain old base 2.

So, if an interviewer asks you to convert base-2 into decimal, ask: "is that in two's complement or not?"

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